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      "#Feature Selection\n",
      "\n",
      "##Subset Selection\n",
      "\n",
      "<p>Probably the most common approach to explicit feature selection is to choose a subset of $k$ features from the total set of $p$ features. The idea scenario would be to consider every possible subset of size $k$ and choose the one that has the best out-of-sample error. Although ideal in theory, in practice, there are $\\binom{p}{k} = \\frac{p!}{k!(p-k)!}$ possible subsets of size $k$. Testing each one is often cost prohibitive when $p$ gets large.<br><br>\n",
      "\n",
      "<b><u>Forward Stepwise Selection</u></b><br>\n",
      "Forward stepwise selection uses a greedy procedure to learn a good subset of $k$ features (though not guaranteed to be the \"best\" subset). The procedure goes as follows:<br>\n",
      "<ul>\n",
      "    <li>1. Initialize: Curr_Best_Subset = { } (i.e., just the intercept)</li>\n",
      "    <li>2. Loop through each feature $j$:</li>\n",
      "        <ul>\n",
      "            <li> a. Add the feature to the current best subset, train a model</li>\n",
      "            <li> b. Get out-of-sample error on the model trained with Curr_Best_Subset+feature $j$</li>\n",
      "        </ul>\n",
      "    <li>3. Choose the feature with the best out-of-sample improvement in error </li>\n",
      "    <li>4. Add this best feature to Curr_Best_Subset, and log the out-of-sample error</li>\n",
      "    <li>5. Repeat steps 2 - 4 until stopping criterion is met.</li>\n",
      "</ul><br>\n",
      "Some possible stopping criteria are:<br>\n",
      "<ul>\n",
      "    <li>Stop at a pre-defined $k$</li>\n",
      "    <li>Stop when (Performance: Best $k$ - Performance: Best $k-1$)$<\\delta$ </li>\n",
      "    <li>Same as above, but using a 1-std error rule or a t-test to compare best $k$ vs. best $k-1$</li>\n",
      "</ul><br>\n",
      "It might not always make sense or be optimal to choose $k$ in advance. The other two stopping criteria amount to continually adding features until the feature does not improve the model, either in an absolute sense or using some statistical test to compare the difference between cross-validated means.<br>\n",
      "\n",
      "\n",
      "\n",
      "The following code shows how this can be done. We'll use the churn prediction data. For a metric, let's assume we care the most about having a good estimate of $P(Y|X)$, so we'll use logistic regression with log-loss as our feature selection criterion. Again, the log-loss is defined as:<br><br>\n",
      "<center>$logloss = -\\frac{1}{n} \\sum\\limits_{i=1}^n \\: y_i*log(p_i)+(1-y_i)*log(1-p_i)$</center><br>\n",
      "We'll also use the last stopping criteria shown above.\n",
      "\n",
      "</p>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#Load the data. Not much prep is need since its a clean dataset\n",
      "import numpy as np\n",
      "import math\n",
      "import pandas as pd\n",
      "import matplotlib.pyplot as plt\n",
      "import course_utils as bd\n",
      "from sklearn import linear_model\n",
      "\n",
      "\n",
      "f = '/Users/briand/Desktop/ds course/hw/Cell2Cell_data.csv'\n",
      "dat=pd.read_csv(f, header=0, sep=',')\n",
      "train, test = bd.trainTest(dat, 0.5)\n",
      "lab = 'churndep'"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 25
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sklearn.cross_validation import KFold\n",
      "\n",
      "\n",
      "def runXVal_LogLoss(cv, X_sub, Y_tr):\n",
      "    '''\n",
      "    Runs LR cross validation with no regularization, returns mean and standard error of mean\n",
      "    '''\n",
      "    ll = []; \n",
      "    for train_index, test_index in cv:\n",
      "        X_tr_f = X_sub.iloc[train_index]\n",
      "        X_va_f = X_sub.iloc[test_index]\n",
      "        Y_tr_f = Y_tr.iloc[train_index]\n",
      "        Y_va_f = Y_tr.iloc[test_index]\n",
      "\n",
      "        lr = linear_model.LogisticRegression()\n",
      "        lr.fit(X_tr_f, Y_tr_f)\n",
      "        P = lr.predict_proba(X_va_f)[:,1]\n",
      "        \n",
      "        ll.append(-1*(((Y_va_f==1)*np.log(P)+(Y_va_f==0)*np.log(1-P)).mean()))\n",
      "\n",
      "    return [np.array(ll).mean(), np.array(ll).std()/np.sqrt(len(ll))]\n",
      "        \n",
      "def LrForward_LogLoss(X_tr, Y_tr, cv):\n",
      "    '''\n",
      "    Runs cross-validated stepwise selection\n",
      "    Does not pick the best features, but returns data\n",
      "    Returns a dictionary that shows at each k: [feature set], x-validated mean, x-validated var \n",
      "    For each loop, chooses the feature with best mean+1stderr\n",
      "    '''\n",
      "    results = {}\n",
      "    curr_best = set([])\n",
      "    cand_list = set(X_tr.columns.values)\n",
      "    k = 1\n",
      "    \n",
      "    while (len(cand_list)>0):\n",
      "        best_mu = 10**10; best_serr = 10**10; \n",
      "        for f in cand_list:\n",
      "            use_x = list(curr_best)+[f]\n",
      "            mu, serr = runXVal_LogLoss(cv, X_tr[use_x], Y_tr)\n",
      "            if ((mu + serr) < (best_mu + best_serr)):\n",
      "                best_mu = mu\n",
      "                best_serr = serr\n",
      "                best_f = f\n",
      "        curr_best.add(best_f) #Add the best feature to the curr_best_set\n",
      "        cand_list = cand_list.difference(curr_best) #Remove the best feature from the candidate set\n",
      "        results[k] = [list(curr_best), best_mu, best_serr]\n",
      "        k+=1\n",
      "        \n",
      "    return results\n",
      "            "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 29
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#Run the forward selection\n",
      "cv = KFold(n=train.shape[0], n_folds = 10)\n",
      "r = LrForward_LogLoss(train.drop(lab,1), train[lab], cv)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 30
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<p>\n",
      "\n",
      "\n",
      "</p>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#Now plot the incremental results\n",
      "ks = []; mus = []; serrs = [];\n",
      "for i in range(len(r.keys())):\n",
      "    ks.append(i+1)\n",
      "    mus.append(r[i+1][1])\n",
      "    serrs.append(r[i+1][2])\n",
      "\n",
      "    \n",
      "best_1serr = min(np.array(mus) + np.array(serrs))\n",
      "plt.clf()\n",
      "plt.plot(ks, mus, 'b.-', label = 'LL of Set k')\n",
      "plt.plot(ks, np.array(mus) + np.array(serrs), 'k+-')\n",
      "plt.plot(ks, np.array(mus) - np.array(serrs), 'k--')\n",
      "plt.plot(ks, np.ones(len(ks))*best_1serr, 'r', label ='1 std err rule')\n",
      "\n",
      "plt.xlim([1,11])\n",
      "\n",
      "plt.title('Stepwise Forward Feature Selection')\n",
      "plt.xlabel('Greedily Selected Subset of Size k')\n",
      "plt.ylabel('X Validated LogLoss')\n",
      "    \n",
      "plt.legend(loc=1, ncol=2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
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       "output_type": "pyout",
       "prompt_number": 31,
       "text": [
        "<matplotlib.legend.Legend at 0x10ab749d0>"
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       "output_type": "display_data",
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VXtV6gDtf/fr16dOnD19++eWTDkWSJKlKqVYtC4CjR48ycuRI4uPjUSqrVa6U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\niSfceRY29gAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10abacb50>"
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<p>The above plot will support any of the stopping criteria defined above. Assuming we haven't predefined $k$, we might choose $k=7$, as that is where the error really seems to plateau. Additionally, we could be more conservative. The red line shows the min value of (mean+1 std error). Any value of $k$ that is less than this value is statistically the same as the $k$ with the lowest mean error. If we were using that rule, we'd potentially stop at $k=3$.<br><br>\n",
      "</p>\n",
      "\n",
      "##SVD Based Dimensionality Reduction\n",
      "\n",
      "<p>Another way to bring the learning down to $k$ features is to find a projection matrix that produces a rank-$k$ approximation of our training data $X$. The best way to get a low-rank approximation, from a signal preservation perspective, is by the use of the Singular Value Decomposition. A review of the SVD can be found here:<br><br>\n",
      "http://nbviewer.ipython.org/github/briandalessandro/DataScienceCourse/blob/master/ipython/Lecture3_PhotoSVD.ipynb\n",
      "<br><br>\n",
      "We'll put it to work here by doing the following:\n",
      "<ul>\n",
      "    <li>Decomposing our training data</li>\n",
      "    <li>For each rank-k approximation of X, get cross-validated error</li>\n",
      "    <li>Compare this error to the Stepwise Forward Feature Selection above</li></ul><br>\n",
      "The scale of the features influences the spectrum of the singular values, so we'll normalize the feature so they all have the same scale.\n",
      "</p>    "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import math\n",
      "import pandas as pd\n",
      "import matplotlib.pyplot as plt\n",
      "import course_utils as bd\n",
      "from sklearn import linear_model\n",
      "from sklearn.preprocessing import scale\n",
      "reload(bd)\n",
      "\n",
      "\n",
      "f = '/Users/briand/Desktop/ds course/hw/Cell2Cell_data.csv'\n",
      "\n",
      "X_train = train.drop(lab, 1)\n",
      "X_test = test.drop(lab, 1)\n",
      "Y_train = train[lab]\n",
      "Y_test = test[lab]\n",
      "\n",
      "X_train_norm = pd.DataFrame(scale(X_train, axis=0, with_mean=True, with_std=True, copy=True), columns = X_train.columns.values)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 32
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<p>Now let's decompose the training data.</p>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "U, sig, Vt = np.linalg.svd(X_train_norm)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 33
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<p>Out of curiosity, let's plot the spectrum to get a sense of how independent the features are."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(np.cumsum(sig**2)/np.sum(sig**2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 34,
       "text": [
        "[<matplotlib.lines.Line2D at 0x109da4210>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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tQibU3W7429/g/ffh2GPtrkZEJDyFRKj/978wfryZ7dKli93ViIiEL9tvlNbV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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x109c91410>"
       ]
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<p>We can see above that the first 5 or 6 singular values account for 70-80% of the sum of squares of our data. Now let's see how a rank-$k$ approximation does in cross-validation.\n",
      "</p>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def LrRankK_LogLoss(U, sig, Vt, Y_tr, cv):\n",
      "    '''\n",
      "    Runs cross-validation on each rank k approximation\n",
      "    '''\n",
      "    results = {}\n",
      "    \n",
      "    for k in range(len(sig)):\n",
      "        X_k = pd.DataFrame(np.matrix(U[:, :k+1]) * np.diag(sig[:k+1]))\n",
      "        mu, serr = runXVal_LogLoss(cv, X_k, Y_tr)\n",
      "        results[k+1] = [mu, serr]\n",
      "        \n",
      "        \n",
      "    return results"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 35
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "r_svd = LrRankK_LogLoss(U, sig, Vt, Y_train, cv)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 36
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<p>Now let's plot the results. We'll also plot it with the stepwise forward selection results above.\n",
      "</p>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.clf()\n",
      "\n",
      "k_svd = []\n",
      "mu_svd = []\n",
      "serr_svd = []\n",
      "\n",
      "for i in range(len(r_svd.keys())):\n",
      "    k_svd.append(i+1)\n",
      "    mu_svd.append(r_svd[i+1][0])\n",
      "    serr_svd.append(r_svd[i+1][1])\n",
      "    \n",
      "plt.plot(k_svd, mu_svd, 'b.-', label = 'SVD Rank-k Proj')\n",
      "plt.plot(k_svd, np.array(mu_svd)+np.array(serr_svd), 'b+')\n",
      "plt.plot(k_svd, np.array(mu_svd)-np.array(serr_svd), 'b--')\n",
      "\n",
      "\n",
      "plt.plot(ks, mus, 'r.-', label = 'Stepwise Forward')\n",
      "plt.plot(ks, np.array(mus) + np.array(serrs), 'r+-')\n",
      "plt.plot(ks, np.array(mus) - np.array(serrs), 'r--')\n",
      "\n",
      "plt.legend(loc=1, ncol=2)\n",
      "\n",
      "plt.title('Stepwise Forward Feature Selection vs SVD Dim Reduction')\n",
      "plt.xlabel('k')\n",
      "plt.ylabel('X Validated LogLoss')\n",
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 41,
       "text": [
        "<matplotlib.text.Text at 0x109935910>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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z7N27F9OnT8fdu3eLLK+mxnnbtm3YsWMHwsPD8fDhQ2RmZhaqs+fOncO9e/fw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\nW6zKbNiwARMmTIBGowEA+Pv7AwCaNGkCLy8vaLVaGAwGaLVaBAQEAAC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       "text": [
        "<matplotlib.figure.Figure at 0x1096be4d0>"
       ]
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<p>This is a very interesting plot. We can see that SVD approach does almost universally worse than the stepwise approach. Its not until we get to the last two features do we see them equal each other. <br><br>\n",
      "My interpretation of this is that the SVD approach does not reduce the feature space with the outcome or learning task in mind. It is purely untangling the covariance structure of the matrix $X$. It is quite possible (based on the evidence above), that the most predictive features corresponded to the lowest singular values. That is why we don't see a large reduction in the cross-validated error until the last few columns are added into the matrix.<br><br>\n",
      "\n",
      "This is not to say or prove that dimensionality reduction with SVD is a generally inferior approach. Because the matrix $X$ is dense and not very high dimensional, the SVD approach just might not be appropriate here. It is also slower, as the cost of generating the SVD isn't always cheap.\n",
      "\n",
      "\n",
      "</p>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}